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Question Number 66968    Answers: 0   Comments: 0

Question Number 66890    Answers: 3   Comments: 0

$$\: \\ $$$$\:\:\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{x}}} \:+\:\boldsymbol{\mathrm{x}}^{\mathrm{2}\boldsymbol{\mathrm{x}}} \:=\:\mathrm{20} \\ $$$$\: \\ $$$$\:\:\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{x}}} \:=\:? \\ $$$$\: \\ $$

Question Number 66875    Answers: 0   Comments: 4

Question Number 66866    Answers: 1   Comments: 1

Question Number 66865    Answers: 1   Comments: 1

$${A}\:{and}\:{B}\:{are}\:{two}\:{towns}\:\mathrm{360}{km}\:{apart}. \\ $$$${An}\:{express}\:{bus}\:{departs}\:{from}\:{A}\:{at} \\ $$$$\mathrm{8}{a}.{m}\:{and}\:{maintains}\:{an}\:{average} \\ $$$${speed}\:{of}\:\mathrm{90}{km}/{h}\:{between}\:{A}\:{and}\:{B}. \\ $$$${Another}\:{bus}\:{starts}\:{from}\:{B}\:{also}\:{at} \\ $$$$\mathrm{8}{a}.{m}\:{and}\:{moves}\:{towards}\:{A}\:{making} \\ $$$${four}\:{stops}\:{at}\:{four}\:{equally}\:{spaced} \\ $$$${points}\:{between}\:{B}\:{and}\:{A}.\:{Each}\:{stop} \\ $$$${is}\:{of}\:{duration}\:\mathrm{5}\:{minutes}\:{and}\:{the} \\ $$$${average}\:{speed}\:{between}\:{any}\:{two}\:{stops} \\ $$$${is}\:\mathrm{60}{km}/{h}.\:{Calculate}\:{the}\:{distance} \\ $$$${between}\:{the}\:{two}\:{buses}\:{at}\:\mathrm{10}{p}.{m}. \\ $$

Question Number 66868    Answers: 0   Comments: 3

$${A}\:{town}\:{N}\:{is}\:\mathrm{340}{km}\:{due}\:{west}\:{of}\: \\ $$$${town}\:{G}\:{and}\:{town}\:{K}\:{is}\:{due}\:{west}\: \\ $$$${of}\:{town}\:{N}.\:{A}\:{helicopter}\:{Zebra}\: \\ $$$${left}\:{G}\:{for}\:{K}\:{at}\:\mathrm{9}{a}.{m}.\:{Another}\: \\ $$$${helicopter}\:{Buffalo}\:{left}\:{N}\:{for}\:{K} \\ $$$${at}\:\mathrm{11}{a}.{m}.\:{Helicopter}\:{Buffalo} \\ $$$${travelled}\:{at}\:{an}\:{average}\:{speed}\:{of}\: \\ $$$$\mathrm{20}{km}/{h}\:{faster}\:{than}\:{helicopter} \\ $$$${Zebra}.\:{If}\:{both}\:{helicopters}\:{reached} \\ $$$${K}\:{at}\:\mathrm{12}.\mathrm{30}{p}.{m},\:{find}\:{the}\:{speed}\:{of}\: \\ $$$${helicopter}\:{Buffalo}. \\ $$

Question Number 66863    Answers: 0   Comments: 0

$$\:\:\:{x}^{\mathrm{3}} \left({b}−{x}\right)^{\mathrm{2}} +{aex}\left({x}−{b}\right)+{e}^{\mathrm{2}} =\mathrm{0}\: \\ $$$${solve}\:{for}\:{x}. \\ $$

Question Number 66862    Answers: 0   Comments: 1

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}3}{x}^{\mathrm{5}} \\ $$

Question Number 66857    Answers: 0   Comments: 3

Question Number 66856    Answers: 1   Comments: 0

$${Two}\:{towns}\:{T}\:{and}\:{S}\:{are}\:\mathrm{300}\:{km}\:{apart}. \\ $$$${Two}\:{buses}\:{A}\:{and}\:{B}\:{started}\:{from} \\ $$$${T}\:{at}\:{the}\:{same}\:{time}\:{travelling}\:{towards} \\ $$$${S}.\:{Bus}\:{B},\:{travelling}\:{at}\:{an}\:{average} \\ $$$${speed}\:{of}\:\mathrm{10}{km}/{h}\:{greater}\:{than}\:{that} \\ $$$${of}\:{A}\:{reached}\:{S}\:\mathrm{1}\frac{\mathrm{1}}{\mathrm{4}}\:{hours}\:{earlier}. \\ $$$$\left({a}\right)\:{Find}\:{the}\:{average}\:{speed}\:{of}\:{A} \\ $$$$\left({b}\right)\:{How}\:{far}\:{was}\:{A}\:{from}\:{T}\:{when} \\ $$$${B}\:{reached}\:{S}. \\ $$

Question Number 66855    Answers: 2   Comments: 0

$${simplify} \\ $$$$\frac{{p}^{\mathrm{2}} +\mathrm{2}{pq}+{q}^{\mathrm{2}} }{{p}^{\mathrm{3}} −{pq}^{\mathrm{2}} +{p}^{\mathrm{2}} {q}−{q}^{\mathrm{3}} } \\ $$

Question Number 66852    Answers: 0   Comments: 3

Question Number 66851    Answers: 2   Comments: 1

Question Number 66849    Answers: 0   Comments: 2

Question Number 66846    Answers: 0   Comments: 2

Question Number 66842    Answers: 0   Comments: 2

Question Number 66845    Answers: 0   Comments: 1

$${A}\:{cylindrical}\:{tank}\:{of}\:{radius}\:\mathrm{2}{m}\: \\ $$$${and}\:{height}\:\mathrm{1}.\mathrm{5}{m}\:{initially}\:{contains} \\ $$$${water}\:{to}\:{a}\:{depth}\:{of}\:\mathrm{50}{cm}.\:{Water} \\ $$$${is}\:{added}\:{to}\:{the}\:{tank}\:{at}\:{the}\:{rate}\:{of}\: \\ $$$$\mathrm{62}.\mathrm{84}{l}\:{per}\:{minute}\:{for}\:\mathrm{15}\:{minutes}. \\ $$$${Find}\:{the}\:{new}\:{height}\:{of}\:{water}\:{in} \\ $$$${the}\:{tank}. \\ $$

Question Number 66833    Answers: 0   Comments: 0

$${Somewhere}\:,\:{there}\:{are}\:{scorpio},\:{snake}\:{and}\:{mouse}. \\ $$$${We}\:{ascertain}\:{that}\:: \\ $$$${Every}\:{morning}\:,\:{each}\:{snake}\:{eats}\:{a}\:{mouse}\:. \\ $$$${Every}\:{afternoon}\:,{each}\:{scorpio}\:\:{kills}\:{a}\:{snake}. \\ $$$${And}\:{every}\:{night}\:,\:{each}\:{mouse}\:{eats}\:{a}\:{scorpio}. \\ $$$${Two}\:{weeks}\:{passed}\:{and}\:{we}\:{find}\:{that}\:{there}\:{was}\:{remaining}\:{only}\:{one}\:{animal}\:. \\ $$$$\left.\mathrm{1}\right)\:{If}\:{that}\:{animal}\:{is}\:{a}\:{snake}\:,\:{how}\:{many}\:{snake}\:{were}\:{there}\:{two}\:{week}\:{ago}? \\ $$$$\left.\mathrm{2}\right){If}\:{that}\:{animal}\:{is}\:{a}\:{scorpio}\:,\:{how}\:{many}\:{scorpio}\:{were}\:{there}\:{two}\:{weeks}\:{ago}? \\ $$$$\left.\mathrm{3}\right){If}\:{that}\:{animal}\:{is}\:{a}\:{mouse}\:,{how}\:{many}\:{mouse}\:{were}\:{there}\:\:{two}\:{weeks}\:{ago}? \\ $$$$\left.\mathrm{4}\right){If}\:{there}\:{are}\:{remaining}\:{one}\:{animal}\:{of}\:{each}\:{breed}\:,\:{How}\:{many}\:{were}\:{they}\:{for}\:{each}\:{breed}. \\ $$

Question Number 66832    Answers: 2   Comments: 0

$${solve}\:{the}\:{system}\:{of}\:{congruence} \\ $$$$\:\:\:\left.\begin{matrix}{{x}\equiv\:\mathrm{1}\:\left({mod}\:\mathrm{5}\right)}\\{{x}\:\equiv\:\mathrm{2}\:\left({mod}\:\mathrm{7}\right)}\\{{x}\equiv\:\:\mathrm{3}\left({mod}\:\mathrm{9}\right)}\\{{x}\:\equiv\:\mathrm{4}\left(\:{mod}\:\mathrm{11}\right)}\end{matrix}\right\} \\ $$

Question Number 66830    Answers: 0   Comments: 4

$${evaluate}. \\ $$$$\:\int_{\mathrm{1}} ^{\:\infty} \:\frac{\mathrm{1}}{{x}^{\mathrm{2}\:} }\:{dx}. \\ $$$$ \\ $$$${can}\:{i}\:{assume}\:\underset{{t}\rightarrow\mathrm{0}} {\:\mathrm{lim}}\:\int_{\mathrm{1}} ^{\:\:{t}} \frac{\mathrm{1}}{{x}^{\mathrm{2}\:} }\:{dx}\:???? \\ $$

Question Number 66827    Answers: 0   Comments: 3

Question Number 66814    Answers: 0   Comments: 0

$${Let}\:{consider}\:{an}\:{integer}\:{serie}\:\left\{{a}_{{n}} {x}^{{n}} \right\}\:{given}\:{by}\:\:{a}_{{n}} \:=\:{H}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{k}}\: \\ $$$$\left.\mathrm{1}\right)\:{Find}\:{out}\:{the}\:{largest}\:{domain}\:{D}\:{of}\:{convergence}\:{of}\:{that}\:{integer}\:{serie} \\ $$$$\left.\mathrm{2}\right)\:\forall\:{x}\in{D}\:\:,\:{explicit}\:{the}\:{sum}\:{S}\left({x}\right)\:{of}\:{the}\:\left\{{a}_{{n}} {x}^{{n}} \right\}\: \\ $$$$\left.\mathrm{3}\right)\:{Calculate}\:\:\int_{−\mathrm{1}} ^{\mathrm{1}} \:{S}\left(\mathrm{1}−{x}\right){S}\left({x}\right)\:{dx}\:. \\ $$$$ \\ $$$$\:\:\:\: \\ $$

Question Number 66803    Answers: 1   Comments: 3

$$\:{prove}\:{that} \\ $$$$\underset{{r}={k}} {\overset{{n}} {\sum}}\:{r}\:=\:\frac{\mathrm{1}}{\mathrm{2}}{n}\left({n}+\mathrm{1}\right) \\ $$$$ \\ $$$${show}\:{with}\:{a}\:{diagram}\:{that}\:{the}\:{volume}\:{of}\:{a}\:{parallepipe}\:{is}\:\:\:{a}.\left({b}×{c}\right) \\ $$

Question Number 66802    Answers: 0   Comments: 6

$${given}\:{that}\: \\ $$$${f}\left({x}\right)\:=\:\mathrm{3}{x}^{\mathrm{3}} \:−\:\mathrm{2}{x}^{\mathrm{2}} \:+\:\mathrm{5}{x}\:+\:\mathrm{7}\:\:{find} \\ $$$$\left.{a}\right)\:\:\alpha\:+\:\beta\:+\:\gamma \\ $$$$\left.{b}\right)\:\alpha\beta\gamma\:\: \\ $$$$\left.{c}\right)\:\alpha^{\mathrm{2}} \:+\:\beta^{\mathrm{2}} \:+\:\gamma^{\mathrm{2}} \\ $$$$\left.{d}\right)\:\alpha^{\mathrm{3}} \:+\:\beta^{\mathrm{3}} \:+\:\gamma^{\mathrm{3}} \\ $$$${any}\:\:{solutions}\:\:{directly}? \\ $$

Question Number 66801    Answers: 0   Comments: 3

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{2}} \sqrt{{x}+{t}^{\mathrm{2}} }{dt}\:\:\:{with}\:{x}\geqslant\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{2}} \:\frac{{dt}}{\sqrt{{x}+{t}^{\mathrm{2}} }} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\left[{of}\:\int_{\mathrm{0}} ^{\mathrm{2}} \sqrt{\mathrm{4}+{t}^{\mathrm{2}} }{dt}\:{and}\:\int_{\mathrm{0}} ^{\mathrm{2}} \frac{{dt}}{\sqrt{\mathrm{3}+{t}^{\mathrm{2}} }}\right. \\ $$$$\left.\mathrm{4}\right)\:{give}\:{g}^{'} \left({x}\right)\:{at}\:{form}\:{of}\:{integral}. \\ $$

Question Number 66800    Answers: 0   Comments: 1

$${calculate}\:{U}_{{n}} =\int_{\frac{\mathrm{1}}{{n}}} ^{{n}} \:\:\:\frac{{arctan}\left({x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$$${and}\:{determine}\:{lim}_{{n}\rightarrow+\infty} {U}_{{n}} \\ $$$$\left.\mathrm{2}\right){find}\:{nature}\:{of}\:\Sigma\:{U}_{{n}} \\ $$

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